Comparison Between Reverberation-Ray Matrix, Reverberation-Transfer Matrix, and Generalized Reverberation Matrix

Abstract

Different matrix formulations have been developed to investigate the elastic-wave propagation in a multilayered solid, which have been widely used in the fields of seismology, ocean acoustics(Pao et al., 2000), and non-destructive evaluation (Lowe, 1995), etc. Transfer matrix (TM) method (Haskell, 1953; Thomson, 1950), as one of the most important matrix formulations, yields a simple configuration and efficient computational ability to facilitate its wide application in many research fields. Stiffness matrix (SM) method (Rokhlin & Wang, 2002; Wang & Rokhlin, 2001) has been proposed to resolve the inherent computational instability for the large product of frequency and thickness in TM method. The SM formulation utilizes the stiffness matrix of each sublayer in a recursive algorithm to obtain the stacked stiffness matrix for the multilayered solid. However, the SM formulation is difficult to identify the generalized-ray propagation in the multilayered solid. In order to evaluate the transient wave propagation in the multilayered solid, Su, Tian, and Pao (Su et al., 2002; Tian & Xie, 2009; Tian et al., 2006) presented reverberation-ray matrix (RRM) formulation. Introducing the local scattering relations at interfaces and the phase relations in sublayers, a system of equations is formulated by a reverberation matrix R , which can be automatically represented as a series of generalized ray group integrals according to the times of reflections and refractions of generalized rays at interfaces. Each generalized ray group integral containing k R represents the set of K times reflections and transmissions of source waves arriving at receivers in the multilayered solid, which is very suitable to automatic computer programming for the simple multilayered-solid configuration. However, the dimension of the reverberation matrix will increase as the number of the sublayers increases, which may yield the lower calculation efficiency of the generalized-ray groups in the complex multilayered solid(Tian & Xie, 2009). In order to increase the calculation efficiency of the generalized-ray groups, Tian presented the reverberation-transfer matrix (RTM) and generalized reverberation matrix (GRM) formulations, respectively. In RTM formulation, RRM formulation is applied to the interested sublayer for the evaluation of the generalized rays and TM formulation to the other sublayers, to construct a RTM of the constant dimension, which is independent of the sublayer number. However, the RTM suffers from the inherent numerical instabilities particularly when the layer thickness becomes large and/ or the frequency is high. GRM